a + 3)(a^3 - 2a^2 + 5a -2)
S k i l l
14 MULTIPLYING OUT THE DISTRIBUTIVE RULE m(a + b) = ma + mb "To multiply a sum, multiply each term of the sum." That is called the distributive rule. m multiplies a, then it multiplies b. We say that m is "distributed" to a and b. Example 1. 2(x + y + z) = 2x + 2y + 2z. We have distributed 2 to x, y, and z. We have "multiplied out." Example 2. 3x4(x² − 5x + 1) = 3x6 − 15x5 + 3x4 That is,
Problem 1. −1(a − b + c − d) What will be the effect of multiplying by −1? To see the answer, pass your mouse over the colored area. Every sign will change. −1(a − b + c − d) = −a + b − c + d Therefore in an equation, we may change all the signs on both sides. This equation
Theoretically, we have multiplied both sides by −1. Problem 2. Multiply out.
f) 2xy(x² − 3xy + y²) = 2x3y − 6x²y² + 2xy3 g) −4xy²(x3y − 6xy² − 2x + 3y + 1) = −4x4y3 + 24x²y4 + 8x²y² − 12xy3 − 4xy² Problem 3. Multiply out and simplify, that is, add the like terms.
A sum by a sum (a + b + c)(x + y + z) First distribute a to x, y, and z. Then distribute b. Then distribute c. (a + b + c)(x + y + z) = ax + ay + az + bx + by + bz + cx + cy + cz Problem 4. Multiply (p − q)(x − y + z). Observe the Rule of Signs (Lesson 4). (p − q)(x − y + z) = px − py + pz − qx + qy − qz Example 3. Multiply out (x − 2)(x + 3). Simplify by adding the like terms. Solution. First distribute x, then distribute −2:
The student should not have to write the first line, but should be able to write the second line -- x² + 3x − 2x − 6 -- immediately. Problem 5. Multiply out. Always simplify by adding the like terms.
Notice: Upon distributing −4, we have anticipated the like terms by aligning them. However, that is not strictly necessary. Problem 6. Multiply out.
Note: The effect of multiplying by x is to increase each exponent by 1; the effect of multiplying by −1 is to change each sign. Please make a donation to keep TheMathPage online. Copyright © 2001-2007 Lawrence Spector Questions or comments? E-mail: themathpage@nyc.rr.com |